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Properties

Label 2-847-11.5-c1-0-50
Degree $2$
Conductor $847$
Sign $-0.780 + 0.625i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)3-s + (1.61 − 1.17i)4-s + (0.927 − 2.85i)5-s + (−0.809 + 0.587i)7-s + (−0.618 − 1.90i)9-s − 2·12-s + (−1.23 − 3.80i)13-s + (−2.42 + 1.76i)15-s + (1.23 − 3.80i)16-s + (−1.85 + 5.70i)17-s + (−1.61 − 1.17i)19-s + (−1.85 − 5.70i)20-s + 21-s + 3·23-s + (−3.23 − 2.35i)25-s + ⋯
L(s)  = 1  + (−0.467 − 0.339i)3-s + (0.809 − 0.587i)4-s + (0.414 − 1.27i)5-s + (−0.305 + 0.222i)7-s + (−0.206 − 0.634i)9-s − 0.577·12-s + (−0.342 − 1.05i)13-s + (−0.626 + 0.455i)15-s + (0.309 − 0.951i)16-s + (−0.449 + 1.38i)17-s + (−0.371 − 0.269i)19-s + (−0.414 − 1.27i)20-s + 0.218·21-s + 0.625·23-s + (−0.647 − 0.470i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.780 + 0.625i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464498 - 1.32180i\)
\(L(\frac12)\) \(\approx\) \(0.464498 - 1.32180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.927 + 2.85i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.23 + 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.85 - 5.70i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.61 + 1.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.89 - 6.46i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.85 + 3.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.28 + 5.29i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.09 - 9.51i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 + (-2.78 + 8.55i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.61 - 1.17i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.09 + 9.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.70 + 11.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02131678266495958060395667620, −8.941159752374250967358847132794, −8.348478712033832747730956121149, −7.01573079876751141363733738715, −6.25266125527130161071067747746, −5.59202909614892378121855660766, −4.80161992551078384595458419716, −3.19478865157651504965247241086, −1.79651625046762537871218811364, −0.68983268978200406714090924973, 2.22100909737956276562279289811, 2.87925569857934023940611611846, 4.12688071217290781634657243456, 5.31633966397534630010666565521, 6.51887970957155090763272677191, 6.88708609974219274471799328849, 7.68206840464541440709195840615, 8.922111594456197934246078415332, 9.945824480622569039926007524019, 10.68645287055453599619263423409

Graph of the $Z$-function along the critical line